To conclude the proof it suffices to show that \(\Psi \)  is dense in \(X \times Y\) .
By [1]} and [2]}, there is a dense open subscheme \(U\subset X\)  such that, for every \(x\)  in \(U(k)\) , the fibre \(Z_x\)  of \(Z\rightarrow X\times Y\rightarrow X\)  over \(x\)  is a normal (not necessarily connected) scheme.
As \(\Sigma \cap U\)  is dense in \(U\) , to show the density of \(\Psi \)  it suffices to prove that, for every \(x\)  in \(U(k)\) , the set \(Y(k) \setminus \bigcup _{i=m+1}^n\pi _{i,x}(Z_{i,x}(k))\)  is dense in \(Y\) .
To do so, it suffices to show that, for \(m+1 \le i\le n\)  and \(x\)  in \(U(k)\) , and for every connected component \(Z^{\prime }\)  of \(Z_{i,x}\) , the restriction of the morphism \(\pi _{i,x}:Z_{i,x}\rightarrow \lbrace x\rbrace  \times Y\cong Y\)  to \(Z^{\prime }\)  is a ramified cover.
